Adherence of Punches as an Example

The calculation of G and for a number of geometries, such as peeling, double cantilever, double torsion, or blister test, can be found in textbooks. We will concentrate on the case of adherence of punches (and especially of a sphere) which is conceptually an important topic via which to understand the connection between adherence, mechanics of contact, and fracture mechanics, or, more simply, what is an area of contact. 

The case of an axisymmetric flat punch of radius a on an elastic half-space was solved by KendalK7) by evaluating the elastic energy = iPb and the potential energy Up = —from the elastic displacement under the load P  

The equilibrium corresponding to G = w is unstable (as for a crack of radius a in an infinite body) and the load P given by G = w is the adherence force at both fixed load and fixed grips. Equation (23) could also be deduced from the value of Kj given by Paris and Sih(8) for a deeply notched bar. Stress and displacemeent at the edge of the contact are, of course, those of fracture mechanics in mode I. 

The case of a sphere of radius R is more subtle, since the contact is no longer conformal and elastic energy is stored under the action of molecular forces. To evaluate it, Johnson, Kendall, and Roberts( ) (JKR theory) first apply the Hertzian load 

(27) the bracketed term represents the correction to the Hertz theory for solids with surface energy. Under zero load the radius of contact is given by 

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